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Chapter 8: Problem 18

Use the formula \(A=\frac{1}{2} a P\) to find the area of the regular polygon described. Find the area of a regular pentagon with an apothem of length \(a=6.5\) in. and each side of length \(s=9.4\) in.

Short Answer

Expert verified
The area is 152.75 square inches.

Step by step solution

01

Understand the Formula and Parameters

The formula for the area of a regular polygon is given by \(A = \frac{1}{2} a P\), where \(A\) is the area, \(a\) is the apothem, and \(P\) is the perimeter of the polygon. We need to find the area using the apothem length \(a = 6.5\) inches and the side length \(s = 9.4\) inches.
02

Calculate the Perimeter

Since the polygon is a regular pentagon, it has 5 equal sides. The perimeter \(P\) can be calculated by multiplying the length of one side \(s\) by the number of sides \(n = 5\). Therefore, \(P = 5 \times 9.4 = 47\) inches.
03

Substitute in the Area Formula

Now that we know both the apothem \(a\) and the perimeter \(P\), we can substitute these into the area formula: \(A = \frac{1}{2} \times 6.5 \times 47\).
04

Calculate the Area

Perform the multiplication in the area formula: First, calculate the product inside, \(6.5 \times 47 = 305.5\). Then, multiply by \(\frac{1}{2}\), so \(A = \frac{1}{2} \times 305.5 = 152.75\).
05

State the Final Result

The area of the regular pentagon is 152.75 square inches.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Regular Polygon
A regular polygon is a multi-sided, closed shape where all sides are of equal length, and all interior angles are equal. This symmetry in a regular polygon ensures that it can be inscribed within a circle. Some common examples include regular triangles, squares, and pentagons.
  • All sides are congruent (the same length).
  • All interior angles are equal.
  • Each vertex lies on the circumference of a circle that encloses the polygon.
Understanding what a regular polygon is helps in grasping why calculations such as area rely on the uniformity of sides and angles. When dealing with such polygons, formulas become quite predictable due to the symmetry involved.
Area Calculation
Finding the area of a regular polygon involves using specific formulas because of its symmetry. The formula used in the exercise, \(A = \frac{1}{2} a P\), is particularly useful:
  • \(A\) stands for the area of the polygon.
  • \(a\) is the apothem, the line from the center to the midpoint of one of the sides.
  • \(P\) is the perimeter which is the total length around the polygon.
The formula is derived from the concept that a regular polygon can be divided into identical isosceles triangles. The area of each triangle can be calculated and summed up in a straightforward manner thanks to this arrangement. Thus, understanding how perimeter and apothem play into this calculation is essential.
Pentagon
A pentagon is a five-sided polygon, and a regular pentagon means each side and angle is equal. This regularity allows us to fit the pentagon into easy calculations, just like in the provided exercise.
  • Five sides, each of equal length.
  • Five angles, each measuring 108 degrees.
  • A regular pentagon can be divided into five equal isosceles triangles.
In geometry, the regular pentagon is often encountered in various applications, from simple tiling problems to complex tessellations. Calculating the area becomes easier as the simplicity of its structure allows for the effective application of geometric formulas.
Apothem
The apothem of a regular polygon is a line from the center of the polygon perpendicular to one of its sides. It has a crucial role in area calculation.
  • Serves as the height of the isosceles triangles formed when dividing the polygon.
  • It is always perpendicular to the center of the side it meets.
  • Key in transforming a polygon into simpler, calculable elements like triangles.
For a regular polygon, the length of the apothem is directly related to both the side length and the number of sides. It鈥檚 instrumental in fully utilizing the formula \(A = \frac{1}{2} a P\), enabling us to find out the polygon鈥檚 total area efficiently.

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Most popular questions from this chapter

The perimeter of a rectangle is \(32 \mathrm{cm} .\) Where \(x\) is the width and \(y\) is the length, express the area \(A\) of the rectangle in terms only of \(x .\)

At the Pizza Dude restaurant, pizza is sold by the slice. If the pizza is cut into 6 pieces, then the selling price is 1.95 dollar per slice. If the pizza is cut into 8 pieces, then each slice is sold for 1.50 dollar In which way will the Pizza Dude restaurant clear more money from sales?

Use the formula \(A=\frac{1}{2} a P\) to find the area of the regular polygon described. Find the area of a regular pentagon with an apothem of length \(a=5.2 \mathrm{cm}\) and each side of length \(s=7.5 \mathrm{cm}\)

Given @ \(O\) with radii \(\overline{O A}\) and \(\overline{O B}\) and chord \(\overline{A B}\). a) What type of figure (sector or segment) is bounded by \(\overline{A B}\) and \(\overline{A B} ?\) b) If \(A B=9.7 \mathrm{cm}\) and \(\ell \overline{A B}=11.4 \mathrm{cm},\) find the perimeter of the figure in (a).

Use the formula \(A=\frac{1}{2} a P\) to find the area of the regular polygon described. In a regular octagon, the measure of each apothem is \(4 \mathrm{cm}\) and each side measures exactly \(8(\sqrt{2}-1) \mathrm{cm} .\) Find the exact area of this regular polygon.
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